Efficiency of test for independence after Box-Cox transformation

We consider the efficiency and the power of the normal theory test for independence after a Box-Cox transformation. We obtain an expression for the correlation between the variates after a Box-Cox transformation in terms of the correlation on the normal scale. We discuss the efficiency of test of independence after a Box-Cox transformation and show that for the family considered it is always more efficient to conduct the test of independence based on Pearson correlation coefficient after transformation to normality. Power of test of independence before and after a Box-Cox transformation is studied for a finite sample size using Monte Carlo simulation. Our results show that we can increase the power of the normal-theory test for independence after estimating the transformation parameter from the data. The procedure has application for generating non-negative random variables with prescribed correlation.     

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

Some implications of the union-intersection principle for tests of sphericity

The sphericity hypothesis may be expressed as an intersection of simpler hypotheses on the invariant subspaces of the variance matrix. Applying the union-intersection principle to dissections of this type establishes a link between tests of independence and tests of sphericity. We use some recent results of Bloomfield and Watson [2] and Knott [4] to derive a class of union-intersection tests for sphericity from likelihood ratio tests of independence of two sets of variates. As well, we show that the ordinary likelihood ratio test for sphericity has a natural union-intersection interpretation.     

On the monotonicity of the power functions of tests based on traces of multivariate beta matrices

It is shown that for the MANOVA problem the power function of the test based on the trace of a multivariate beta matrix is monotonically increasing in each noncentrality parameter provided that the cutoff point is not too large. This result is also true for the problem of testing independence of two sets of variates.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

Maximum eccentricity as a union-intersection test statistic in multivariate analysis

Union-intersection is a heuristic method of test construction developed by S. N. Roy. Among the well-known applications of this principle is the test for independence between two sets of variates which leads directly to the concept of canonical correlation. Another multivariate application of considerable importance is the test of internal independence. In this article we consider the structure of a correlation matrix and derive a union-intersection test statistic for internal independence. This statistic will be shown to be a function of the maximum eccentricity of the p-dimensional correlation ellipsoid x′R^?1x = 1. The statistic will be applied to problems in factor analysis and categorical scaling.     

General canonical correlations with applications to group symmetry models

In this paper, we define general canonical correlations, which generalize the canonical correlations developed by Hotelling, and general canonical covariate pairs, the corresponding linear statistic. We also define canonical variance distances with corresponding canonical distance variates. In a rather broad setting, these parameters and their corresponding linear statistics are characterized in terms of certain eigenvalues and eigenvectors. For seven of the ten group symmetry testing problems discussed in Andersson, Brøns, and Jensen (1983) [4], these are the eigenvalues used to represent the maximal invariant statistic, and additional observations regarding the canonical correlations are made for these testing problems.     

Estimation and inference for dependence in multivariate data

In this paper, a new measure of dependence is proposed. Our approach is based on transforming univariate data to the space where the marginal distributions are normally distributed and then, using the inverse transformation to obtain the distribution function in the original space. The pseudo-maximum likelihood method and the two-stage maximum likelihood approach are used to estimate the unknown parameters. It is shown that the estimated parameters are asymptotical normally distributed in both cases. Inference procedures for testing the independence are also studied.     

Exact nonnull distribution of Wilks’ statistic: The ratio and product of independent components

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks’ statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

Construction of asymmetric multivariate copulas

In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates.     

Extensions of Wilks' integral equations and distributions of test statistics

Summary Wilks [26] introduced two integral equations in connection with distribution problems in statistics. He called them Type A and Type B equations. Tretter and Walster ([22], [24]) solved the Type B equation and obtained the null and non-null distributions of the likelihood ratio criterion for testing linear hypotheses in the multinormal case. In this article we present several types of solutions of these equations along with new equations called Types C, D, E and F with their solutions. These include the integral equations satisfied by the density of a random variable which is (a) product of independent real gamma variates; (b) products of independent real beta variates; (c) ratio of products of independent beta and gamma variates; (d) arbitrary powers of products of gamma and beta variates; (e) arbitrary powers of products and ratios of beta and gamma variates, and more general cases.     

On the distributions of the latent roots and traces of certain random matrices

It is shown that differential equations given by the author may be used recursively to construct certain multivariate null distributions in reduced form. These include the distributions of individual latent roots of B = S₁(S₁ + S₂)⁻¹, and distributions of Tr B and Tr S₁S₂⁻¹, for small numbers of variates.     

Asymptotic Bayesian structure learning using graph supports for Gaussian graphical models

The theory of Gaussian graphical models is a powerful tool for independence analysis between continuous variables. In this framework, various methods have been conceived to infer independence relations from data samples. However, most of them result in stepwise, deterministic, descent algorithms that are inadequate for solving this issue. More recent developments have focused on stochastic procedures, yet they all base their research on strong a priori knowledge and are unable to perform model selection among the set of all possible models. Moreover, convergence of the corresponding algorithms is slow, precluding applications on a large scale. In this paper, we propose a novel Bayesian strategy to deal with structure learning. Relating graphs to their supports, we convert the problem of model selection into that of parameter estimation. Use of non-informative priors and asymptotic results yield a posterior probability for independence graph supports in closed form. Gibbs sampling is then applied to approximate the full joint posterior density. We finally give three examples of structure learning, one from synthetic data, and the two others from real data.     

Nonparametric tests for conditional independence in two-way contingency tables

Testing for the independence between two categorical variables R and S forming a contingency table is a well-known problem: the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual a set of p characteristics is also observed. Those explanatory variables, likely to be associated with R and S, can play a major role in their possible association, and it can therefore be interesting to test the independence between R and S conditionally on them. In this paper, we propose two nonparametric tests which generalise the chi-square and the likelihood ratio ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics under the conditional independence hypothesis is derived; the finite sample behaviour of the procedure is analysed through some Monte Carlo experiments and the approach is illustrated with a real data example.     

Tests for independence in non-parametric heteroscedastic regression models

Consistent procedures are constructed for testing independence between the regressor and the error in non-parametric regression models. The tests are based on the Fourier formulation of independence, and utilize the joint and the marginal empirical characteristic functions of the regressor and of estimated residuals. The asymptotic null distribution as well as the behavior of the test statistic under alternatives is investigated. A simulation study compares bootstrap versions of the proposed tests to corresponding procedures utilizing the empirical distribution function.     

Exact robustness studies of the test of independence based on four multivariate criteria and their distribution problems under violations

Summary Exact robustness studies against non-normality have been carried out for test of independence based on the four multivariate criteria: Hotelling's trace,U ^(p) , Pillai's trace,V ^(p) , Wilks' criterion,W ^(p) , and Roy's largest root,L _(p) . The density functions ofU ^(p) ,W ^(p) andL _(p) have been obtained in the canonical correlation case and further the moments ofU ^(p) and m.g.f. ofV ^(p) have been derived. All of the study is based on Pillai's distribution of the characteristic roots under violations. Numerical results for the power function have been tabulated for the two-roots case. Slight non-normality does not affect the independence test seriously.V ⁽²⁾ is found to be most robust against nonnormality. Yu-Sheng Hsu is now with Georgia State University, Atlanta.     

The Matsumoto-Yor property and the structure of the Wishart distribution

This paper establishes a link between a generalized matrix Matsumoto-Yor (MY) property and the Wishart distribution. This link highlights certain conditional independence properties within blocks of the Wishart and leads to a new characterization of the Wishart distribution similar to the one recently obtained by Geiger and Heckerman but involving independences for only three pairs of block partitionings of the random matrix.In the process, we obtain two other main results. The first one is an extension of the MY independence property to random matrices of different dimensions. The second result is its converse. It extends previous characterizations of the matrix generalized inverse Gaussian and Wishart seen as a couple of distributions.We present two proofs for the generalized MY property. The first proof relies on a new version of Herz's identity for Bessel functions of matrix arguments. The second proof uses a representation of the MY property through the structure of the Wishart.     

Asymptotically optimal tests for parametric functions against ordered functional alternatives

There are hypothesis testing problems for (nonlinear) functions of parameters against functional ordered alternatives for which a reduction to a conventional order-restricted hypothesis testing problem may not be feasible. While such problems can be handled in an asymptotic setup, among the available choices, it is shown that the union-intersection principle may have certain advantages over the likelihood principle or its ramifications. An application to a genomic model is also considered.     

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R_n,A, where A is a subset of {1,…,p} of cardinality |A|>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.